For larger \(k\) , less is known. For \(k\)  between \(n\)  and \(n^2\) , Chvátal showed that \(2n\log _2 k + 4n\)  guesses suffice. For any \(k\ge n\) , this was improved to \(2n \log _2 n +2n+\lceil k/n\rceil + 2\)  by Chen, Cunha, and Homer [1]}, further to \(n\lceil \log _2 k\rceil + \lceil (2-1/k)n\rceil + k\)  by Goodrich [2]}, and again to \(n\lceil \log _2 n\rceil -n+k+1\)  by Jäger and Peczarski [3]}. As a comparison, we note that if \(k=k(n)\)  is polynomial in \(n\) , then the entropy lower bound is simply \(\Omega (n)\) . This gap is a very natural one as Doerr, Doerr, Spöhel, and Thomas [4]} showed in a relatively recent paper that if one uses a non-adaptive strategy, there is in fact a lower bound of \(\Omega (n \log n)\)  when \(k = n\) . In the same paper they also use an adaptive strategy to significantly narrow this gap, showing that \(\mathcal {O}(n \log \log n)\)  guesses suffice for \(k=n\) . Moreover, by proving a general relation between \(n=k\)  Mastermind with only black pegs and \(k\ge n\)  Mastermind with both black and white pegs, they show that \(\mathcal {O}(n\log \log n + k/n)\)  guesses suffice for any \(k\ge n\) .
